exact implementation of multiple initial conditions in the dq solution of higher-order odes

The differential quadrature method (dqm) is one of the most elegant and useful approximate methods for solving initial and/or boundary value problems. it is easy to use and also straightforward to implement. however, the conventional dqm is well-known to have some difficulty in implementing multiple initial and/or boundary conditions at a given discrete point. to overcome this difficulty, this paper presents a simple and accurate differential quadrature methodology in which the higher-order initial conditions are exactly implemented. the proposed methodology is very elegant and uses a set of simple polynomials with a simple transformation to incorporate the higher-order initial conditions at the initial discrete time point. the order of accuracy of the proposed method for solving an rth order ordinary differential equation is “m + r – 1,” where m being the number of discrete time points. this is better than the accuracy of the cbcge (direct coupling the boundary/initial conditions with the discrete governing equations) and mwcm (modifying weighting coefficient matrices) approaches whose order is in general “m – 1.” some test problems are also provided to highlight the superiority of the proposed method over the cbcge and mwcm approaches.